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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdmnd | Structured version Visualization version GIF version |
Description: A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdmnd | ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdcmn 31131 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) | |
2 | cmnmnd 19140 | . 2 ⊢ (𝑊 ∈ CMnd → 𝑊 ∈ Mnd) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 Mndcmnd 18127 CMndccmn 19124 SLModcslmd 31126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-cmn 19126 df-slmd 31127 |
This theorem is referenced by: slmdbn0 31134 slmdvacl 31138 slmdass 31139 slmd0vcl 31147 slmd0vlid 31148 slmd0vrid 31149 |
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